NOTE-2: the "inverse function" is so general and complex, because can operates with point, line and polygon inputs. Here we can discuss the simplest case, when the input is like a point, with no area or length.

depends if it is high tide or low tide....
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Mapperz♦Sep 25 '12 at 13:15

Yes ;-) Sorry about the "no practical utility" of my little beach... Well, with a real beach you need also the slope... And a really useful model need a function that transforms time into tide height.
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Peter KraussSep 25 '12 at 18:32

What, precisely, is the question? All we have is a picture. Do you seek some distinguished point within a polygon? If so, what property distinguishes it? E.g., should it (a) be furthest from all exterior points; (b) minimize some average function of distance over the polygon (e.g., the centroid minimizes the squared distance); (c) the center of a minimal bounding circle/rectangle/ellipse/some other shape; (d) minimize some average function of distance to the boundary; (e) the center of a maximal inscribed circle/rectangle/triangle; etc., etc?
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whuber♦Sep 26 '12 at 12:48

Sorry my english, you can edit it... The question is "What is the best estimated mean width, w, for a given reference-shape (ex. square or circle)?", where "best estimation" supply equal areas, the ST_Area(st_buffer(point,bestW,shape)) and the ST_Area(island). My answer below, complements my descriptions... About "best fit" see this complementary question.
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Peter KraussSep 26 '12 at 14:15

It is still very difficult to determine what you are trying to do. The calculations appear to determine the radius of a circle of a specified area and the side of a square of specified area. What GIS problem does that solve?
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whuber♦Sep 26 '12 at 21:11

DEMONSTRATING

The main shapes, circle and square, have areas

A = pi*w^2 # circle with radius w
A = 2*w^2 # square circumscribed into a circle of radius w

The intermediary shapes are the regular polygons of n sides. The square have n=4, the octagon n=8, etc. So, we can substitute the factor of w^2 by a function f(n) that supply the adequate value for the n sides.
The formula of the area of a regular polygon of n sides with a circumradiusw, is:

A = 0.5*n*sin(2*pi/n)*w^2

Where we can check: for n=4 the factor 0.5*n*sin(2*pi/n) is 2; for n=400 (infinite) is 3.14 (~pi).

For the standard ST_Buffer(g,'quad_segs=N') polygon generator we can check that this N is 4*n.